I use to run my TV and charge my computer off a power inverter hooked to a battery. I would charge the battery with my generator.
May the digital gods have mercy on your hardware.
Short answer: NO
Long answer: Only if the genset makes steady 60hz, 110vAC power. Fluctuations are the death of 110vAC to 5/12vDC transformers and hardware on the low voltage side. Surge protectors ONLY work to shunt overvoltage to ground, which with a generator isn't going to work. You could (potentially) get yourself a power filter (not a surge protector) and hook that between your computer and genset. If you go this route I would actually recommend getting an APC BackUPS RS1500 with AVR to automatically protect your computer from both high and low voltage and Hz surge.
Easily 90% of the emergency generator market is made up of non-computer-safe gensets. Most of the portables will not make clean power. The exception will be the inverter generators which are generally dead on and smooth power. The military doesn't use Chinese-built portables, I guarantee you that a mil-spec generator will provide the best, most accurate power available along with EM interference supression. Dead hardware and military ops just don't mix well.
Back during Andrew I ran a old type tv on a generator and it tv worked but the remote wouldn't. Soon at the power came back on the remote worked fine. Why?
...Didn't have any problems...